Introduction all QFTs that we use in physics are in some sense - - PowerPoint PPT Presentation

The TT Deformation of Quantum Field Theory

John Cardy

University of California, Berkeley All Souls College, Oxford

ICMP , Montreal, July 2018

TT deformation

Introduction

all QFTs that we use in physics are in some sense effective field theories, valid over only some range of energy/length scales if they are (perturbatively) renormalizable, this range of scales may be very large and they have more predictive power, but eventually new physics should enter if they are non-renormalizable (the action involves

• perators with dimension > d) they may still be useful up to

some energy scale ∼ UV cut-off Λ even so they may still make sense at higher energies if they have a ‘UV completion’, eg. if they flow from a non-trivial RG fixed point – ‘asymptotic safety’

TT deformation

However another possibility is that the UV limit is not a conventional UV fixed point corresponding to a local QFT, but is something else (eg string theory):

IR UV

?

TT deformation

The TT deformation of 2d QFT is an example of a non-renormalizable deformation of a local QFT for which, however, many physical quantities make sense and are finite and calculable in terms of the data of the undeformed theory. However this deformation is very special – this has been termed ‘asymptotic fragility,’ which could be used as a constraint on physical theories.

TT deformation

What is TT?

consider a sequence of 2d euclidean field theories T (t) (t ∈ R) in a domain endowed with a flat euclidean metric ηij, each with a local stress-energy tensor T(t)

ij (x) ∼ δS(t)/δgij(x)

T (0) is a conventional local QFT (massive, or massless (CFT)) deformation is defined formally by S(t+δt) = S(t) − δt

• det T(t) d2x

equivalently by inserting

• det T(t)d2x into correlation

functions ⇒ note that this uses T(t), not T(0) ⇐

TT deformation

det T = 1

2ǫikǫjlTijTkl ∝ TzzT¯ z¯ z − T2 z¯ z in complex coordinates

for a CFT this is TzzT¯

z¯ z ≡ TT

since this has dimension 4, we would expect det T ∼ Λ4 Zamolodchikov (2004) pointed out that by conservation ∂iTij = 0 that ∂ ∂ym ǫikǫ jlTij(x)Tkl(x + y) = ∂ ∂xi ǫmkǫ jlTij(x)Tkl(x + y) so that in any translationally invariant state ǫikǫ jlTij(x)Tkl(x) = ǫikǫ jlTij(x)Tkl(x + y) finite, and calculable in terms of matrix elements of Tij so the deformation is in some sense ‘solvable’

TT deformation

TT as a topological deformation

In this talk I’ll describe a different approach which also works in non-translationally invariant geometries e(δt/2)

• ǫikǫjlTij(x)Tkl(x)d2x =
• [dhij]e−(1/2δt)
• ǫikǫ jlhij(x)hkl(x)d2x+
• hijTijd2x

hij = O(δt) may be viewed as an infinitesimal change in the metric gij = ηij + hij in 2d we can always write hij = ai,j + aj,i + δijΦ where ai is an infinitesimal diffeomorphism xi → xi + ai(x) and eΦ ∼ 1 + Φ is the conformal factor

TT deformation

however, at the saddle point h = h∗[T] (which is sufficient since the integral is gaussian) Tij ∝ ǫikǫ jlh∗

kl

conservation ∂iTij = 0 then implies that Φ = 0, so the metric is flat Φ can be absorbed into the diffeomorphism moreover we can take ai,j = aj,i

TT deformation

the action is then (2/δt)

• ǫikǫ jl(∂iaj)(∂kal)d2x − 2
• (∂iaj)Tijd2x

= (2/δt)

• ∂i(ǫikǫ jlaj∂kal)d2x − 2
• ∂i(ajTij)d2x

and so is topological: for a simply connected domain only a boundary term for a closed manifold, only contributions from nontrivial windings of ai

• nly hij = 2ai,j needs to be single valued

TT deformation

Torus

L L’ L+L’

torus made by identifying opposite edges of a parallelogram with vertices at (0, L, L′, L + L′) in C saddle point is translationally invariant h∗ij = δt ǫikǫ jlTkl(0) = δt ǫikǫ jl(1/A)

• Lk∂Ll + L′

k∂L′

l

• log Z(t)

(A = L ∧ L′ = area) change in log Z(t) is

• Tij(x)h∗ij[T]cd2x = (δt)ǫikǫ jl

Li∂Lj + L′

i∂L′

j

• Tkl(0)

TT deformation

Evolution equation for partition function ∂ ∂tZ(t)(L, L′) = ǫikǫ jl Lk∂Ll + L′

k∂L′

l

• (1/A)
• Lk∂Ll + L′

k∂L′

l

• Z(t)(L, L′)

In terms of Z(t) ≡ Z(t)/A ∂tZ = (∂L ∧ ∂L′)Z simple linear PDE, first order in ∂t if log Z ∼ −ft A, ∂t ft = −f 2

t

⇒ ft = f0 1 + f0t – no new UV divergences in the vacuum energy

TT deformation

interpretation as a stochastic process

∂tZ = (∂L ∧ ∂L′)Z is of diffusion type, where Z(t) is the pdf for a Brownian motion (Lt, L′

t) in moduli space with

E

• (Lt1 − Lt2) ∧ (L′

t1 − L′ t2)

• = |t1 − t2|

In particular the mean area E[Lt ∧ L′

t] ∼ t as t → +∞, with,

however, absorbing boundary conditions on L ∧ L′ = 0. The relation between the two approaches is Z(t)(L0, L′

0) = E

• Z(0)(Lt, L′

t)

• TT deformation

Finite-size spectrum

L L’ L+L’ H P

Z(L, L′) = Tr e−(R Im τ)ˆ

H(R)+i(R Re τ)ˆ P(R)

=

• n

e−(R Im τ)E(t)

n (R)+i(R Re τ)Pn(R)

where R = |L|, τ = L′/L and ˆ H, ˆ P are the energy and momentum operators for the theory defined on a circle of circumference R. PDE for Z(t) then leads after some algebra to [Zamolodchikov 2004] ∂tE(t)

n (R) = −E(t) n (R)∂RE(t) n (R) − P2 n/R For Pn = 0 this is the inviscid Burgers equation.

TT deformation

If T (0) is a CFT, E(0)

n (R) = 2π ˜

∆n/R where ˜ ∆n = ∆n − c/12 Solution is then (with Pn = 0) E(t)

n (R) = R

2t  1 −

• 1 − 8π ˜

∆n t R2   energies with ˜ ∆n > 0 become singular at some finite t > 0 energies with ˜ ∆n < 0 become singular at some finite t < 0

TT deformation

Thermodynamics

identifying R ≡ β = 1/kT, E(t)

0 (β) = βft(β), where ft = free energy per unit length

for fixed t < 0 there is a transition at finite T ∼ 1/(−ct)1/2 this is of Hagedorn type where the density of states grows exponentially if T (0) is a free boson, then E(t)

n (R, P) is the spectrum of the

Nambu-Goto string [Caselle et al. 2013] which is known to have a Hagedorn transition as a world sheet theory

• n the other hand for t > 0 the free energy is analytic but

the energy density E is finite as T → ∞, suggesting another branch with negative temperature

TT deformation

S-matrix

if T (0) is a massive QFT, the single particle mass spectrum M is not affected by the deformation the 2-particle energies for MR ≫ 1 have the expected form E = 2

• M2 + P2

where however P is quantized according to PR + δ(P) ∈ 2πZ, where δ(t)(P) is the scattering phase shift consistency with the evolution equation then requires δ(t) = δ(0) − t M2 sinh θ where P = M sinh(θ/2) this is equivalent to a CDD factor in the 2-particle S-matrix

[Smirnov-Zamolodchikov 2017]

S(t)(θ) = e−itM2 sinh θ S(0)(θ)

TT deformation

if T (0) is integrable, so is T (t), and applying Thermodynamic Bethe Ansatz to the deformed S-matrix yields the expected form for E(t)

n (R) [Cavaglià et al. 2016]

in fact this dressing of the S-matrix works for non-integrable theories as well: [Dubovsky et al. 2012, 2013] S(t)({p}) = e−i(t/8)

a<b ǫij pi apj b S(0)({p})

it corresponds to the dressing of the original theory by Jackiw-Teitelboim [1985, 1983] (topological) gravity: S(t) = S(gij, matter) + √−g(φR − Λ)d2x where Λ ∼ t−1 the torus partition function of this theory has been computed and shown to satisfy the PDE of the TT deformed theory [Dubovsky et al. 2017, 2018]

TT deformation

Simply connected domain

boundary action is (1/8δt)

• ǫ jl(aj∂kal)dsk − 2
• ǫikajTijdsk − λ
• akdsk

λ = lagrange multiplier enforcing

• akdsk = 0

fermion on boundary: coupling to T simplifies with conformal boundary condition T⊥ = 0 gaussian integration then gives δ log Z ∝ δt

|s−s′|<ℓ/2

G(s − s′)T⊥⊥(s)T⊥⊥(s′)c ǫkl dskdsl where ℓ = perimeter and G(s − s′) = 1

2sgn(s − s′) − (s − s′)/ℓ

TT deformation

for a disk |x| < R, we may decompose into modes a⊥(θ) =

n aneinθ

the n = 0 mode gives the evolution equation ∂tZ = (1/4π)(∂R − 1/R)∂RZ where R = perimeter/2π corresponds to the stochastic (Bessel) process ∂tR = − 1 4πR + η(t) , η(t′)η(t′′) = 1 2πδ(t′ − t′′) curvature driven dynamics as in 2d coarsening

TT deformation

Other directions

in the holographic AdS/CFT correspondence, deforming the boundary CFT with t > 0 has been argued to be equivalent to going into the bulk of AdS3 by a distance O(√t) [McGough et al. 2016] in Minkowski space CFT, adding TT to the action corresponds to soft left-right scattering: t > 0 (resp. < 0) corresponds to attractive (repulsive) interaction and superluminal (subliminal) propagation of light signals in a background with finite energy density [Cardy 2016] TT has also been argued to lead to the formation of shocks in the hydrodynamic effective theory [Bernard-Doyon 2015]

TT deformation

Summary

the TT (more properly, det T) deformation of a local 2d QFT gives a computable example of a nonlocal UV completion

• f an effective field theory

many physical quantities (partition functions, spectrum, S-matrix) are UV finite, but local operators do not appear the make sense it is solvable because in a sense it is topological, and it also appears to be related to a dressing of the theory by (topological) gravity it would be interesting to find similar deformations in higher dimensions

TT deformation